GAUSS’S SHORTCUT

1. Instead of adding 1 + 2 + 3 + 4 . . . + 100, add the numbers in pairs, like this: (1 + 100) + (2 + 99) + (3 + 98) . . . (50 + 51). Each pair adds up to 101, and there are 50 pairs, so the total is 101 x 50, or 5,050.

2. The same method yields 500 pairs (1 + 1000) + (2 + 999) + (3 + 998) . . . + (500 + 501) that each add up to 1,001. So the answer is 500 x 1,001, or 500,500.

3. The sum of the even numbers from 2 to 2,000 is simply twice the sum of all the whole numbers from 1 to 1,000. So the answer is twice 500,500, or 1,001,000.

4. As the diagram demonstrates, the sum of a series of odd numbers starting with 1 is always a square number. To be exact, 1 + 3 + 5 + 7 + (2n – 1) equals the square number n². So the sum of the odd numbers 1 to 1,001 is 500², or 250,000. The diagram comes from the book Proofs Without Words by Roger B. Nelson (Mathematical Association of America).

SEEING DOTS

1. Adjacent groups of dots add up to 10. There are six groups of 10 dots, which equals 60 dots.
2. Ignore the colors. You’ll notice that the figure is a square with nine dots on a side (9 x 9, or 81 dots) minus eight groups of two dots each (16 dots): 81 – 16 = 65 dots.
3. This figure looks like a triangle on top of an inverted triangle. But ignore the colors and focus on the diagonal lines and you’ll see that the form is a squished square with nine dots on a side. So the number of dots is 9 x 9, which equals 81.

ADDING BACKWARD

1.  

2. To compute a 15 percent restaurant tip on a $26.14 bill, first compute 10 percent, which is roughly $2.60. Half of that is $1.30, which is 5 percent of the bill. Add the two numbers together to get the standard 15 percent tip: $2.60 + 1.30 = $3.90. A true stickler would leave $3.92, because 15 percent of $26.14 is actually $3.921; a satisfied, easygoing diner would just round up to $4. A real mensch would leave 20 percent.